Chromatic Number of Triangle-free Graphs With Some Forbidden Subgraphs

Abstract

Gyarfas conjectured that for a given forest F, there exists an integer function f(F, ω(G)) such that χ(G) ≤ f(F, ω(G)) for any F-free graph G ,w hereχ(G )a ndω(G )a re the chromatic number and the clique number of G ,r espectively. The broomB(m, n )i s the tree of order m + n obtained from identifying a vertex of degree 1 of Pm with the center of K1,n .I n this note, we prove that if G is a triangle-free and B(m, n)-free graph, then χ(G) ≤ m + n − 1, and for a given tree T ,i fG is a triangle-free, C4-free and T-free graph, then χ(G) ≤| T |− 1. We consider finite, undirected and simple graphs G with vertex set V (G )a nd edge setE(G). Let |G| denote the number of vertices in G.F or a vertexv ∈ V (G), N (v) denotes the set of neighbors of the vertex v in the graph G ,a ndM (v )= V (G) N (v) ∪{ v} .L etδ(G), Δ(G) denote the minimum, maximum vertex degree of a graph G, respectively. For A ⊆ V (G), G(A) denotes the subgraph induced by A.I fG(A) has no edge, A is called an independent set; if G(A) is a complete graph, A is called a clique. The maximum cardinality of an independent set is called the independence number of G, denoted by α(G); the maximum cardinality of a clique is called the clique number, denoted by ω(G). The chromatic number of G, denoted by χ(G), is the minimum number k such that the vertices of G can be partitioned into k independent sets. The degeneracy of G, denoted by deg(G), is the maximum value of δ(H) for all subgraph H of G, that is, deg(G )=m ax{δ(H ): H ⊆ G} .I t is well known thatω(G) ≤ χ(G) ≤ deg(G )+1 holds for any graph G. By a classical result of Erdos (2) , we know that the difference χ(G) − ω(G) can be arbitrarily large.